Sometimes the easiest way to answer such questions is via simulationinstead of analysis. Here is a Monte Carlo estimate, based on 10^6trials, of the distribution of "Het" when all 12! orderings of the12 sample values are equally likely:

Het Prob 0 .021822 1 0 2 .389284 3 0 4 .519354 5 0 6 .069540

Here is an estimate of the distribution of "L-H Het" and "H-L Het",again based on 10^6 trials: